2004 reasoning in quantum theory sharp and unsharp quantum logics

8/7/2019 2004 Reasoning in Quantum Theory Sharp and Unsharp Quantum Logics

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Maria Luisa Dalla Chiara

Roberto Giuntini

Richard Greechie

REASONING IN QUANTUM

THEORYSharp and Unsharp Quantum Logics


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Contents

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiPART I Mathematical and Physical Background . . . . . . . 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Chapter 1. The mathematical scenario of quantum theory and vonNeumanns axiomatization . . . . . . . . . . . . . . . . . 9

1.1. Algebraic structures . . . . . . . . . . . . . . . . . . . . . . . 91.2. The geometry of quantum theory . . . . . . . . . . . . . . . . 241.3. The axiomatization of orthodox QT . . . . . . . . . . . . . . . 311.4. The logic of the quantum events . . . . . . . . . . . . . . . 341.5. The logico-algebraic approach to QT . . . . . . . . . . . . . . 38

Chapter 2. Abstract axiomatic foundations of sharp QT . . . . . . . 412.1. Mackeys minimal axiomatization of QT . . . . . . . . . . . . 422.2. Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.3. Event-state systems . . . . . . . . . . . . . . . . . . . . . . . . 522.4. Event-state systems and preclusivity spaces . . . . . . . . . . 55

Chapter 3. Back to Hilbert space . . . . . . . . . . . . . . . . . . . . 653.1. Events as closed subspaces . . . . . . . . . . . . . . . . . . . . 653.2. Events as projections . . . . . . . . . . . . . . . . . . . . . . . 673.3. Hilbert event-state systems . . . . . . . . . . . . . . . . . . . 683.4. From abstract orthoposets of events to Hilbert lattices . . . . 70

Chapter 4. The emergence of fuzzy events in Hilbert space quantumtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.1. The notion of effect . . . . . . . . . . . . . . . . . . . . . . . . 754.2. Effect-Brouwer Zadeh posets . . . . . . . . . . . . . . . . . . . 774.3. Mac Neille completions . . . . . . . . . . . . . . . . . . . . . . 814.4. Unsharp preclusivity spaces . . . . . . . . . . . . . . . . . . . 82

Chapter 5. Effect algebras and quantum MV algebras . . . . . . . . 87

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5.1. Effect algebras and Brouwer Zadeh effect algebras . . . . . . . 875.2. The Lukasiewicz operations . . . . . . . . . . . . . . . . . . . 94

5.3. MV algebras and QMV algebras . . . . . . . . . . . . . . . . . 965.4. Quasi-linear QMV algebras and effect algebras . . . . . . . . . 107

Chapter 6. Abstract axiomatic foundations of unsharp quantum theory1156.1. A minimal axiomatization of unsharp QT . . . . . . . . . . . 1156.2. The algebraic structure of abstract effects . . . . . . . . . . . 1196.3. The sharply dominating principle . . . . . . . . . . . . . . . . 1246.4. Abstract unsharp preclusivity spaces . . . . . . . . . . . . . . 1276.5. Sharp and unsharp abstract quantum theory . . . . . . . . . . 131

Chapter 7. To what extent is quantum ambiguity ambiguous? . . . . 1377.1. Algebraic notions of sharp . . . . . . . . . . . . . . . . . . . 1 377.2. Probabilistic definitions of sharpness . . . . . . . . . . . . . 142

PART II Quantum Logics as Logic . . . . . . . . . . . . . . . . 147

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Chapter 8. Sharp quantum logics . . . . . . . . . . . . . . . . . . . . 1558.1. Algebraic and Kripkean semantics for sharp quantum logics . 1558.2. Algebraic and Kripkean realizations of Hilbert event-state

systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1628.3. The implication problem in quantum logic . . . . . . . . . . . 1648.4. Five polynomial conditionals . . . . . . . . . . . . . . . . . . . 1 658.5. The quantum logical conditional as a counterfactual conditional1678.6. Implication-connectives . . . . . . . . . . . . . . . . . . . . . . 168

Chapter 9. Metalogical properties and anomalies of quantum logic . 1719.1. The failure of the Lindenbaum property . . . . . . . . . . . . 1719.2. A modal interpretation of sharp quantum logics . . . . . . . . 174

Chapter 10. An axiomatization ofOL and OQL . . . . . . . . . . . 17910.1. The calculi for OL and OQL . . . . . . . . . . . . . . . . . . 17910.2. The soundness and completeness theorems . . . . . . . . . . 181

Chapter 11. The metalogical intractability of orthomodularity . . . . 18511.1. Orthomodularity is not elementary . . . . . . . . . . . . . . 18611.2. The embeddability problem . . . . . . . . . . . . . . . . . . . 1 8811.3. Hilbert quantum logic and the orthomodular law . . . . . . . 189

Chapter 12. First-order quantum logics and quantum set theories . . 19312.1. First-order semantics . . . . . . . . . . . . . . . . . . . . . . 19312.2. Quantum set theories . . . . . . . . . . . . . . . . . . . . . . 198

Chapter 13. Partial classical logic, the Lindenbaum property and thehidden variable problem . . . . . . . . . . . . . . . . . . 201

13.1. Partial classical logic . . . . . . . . . . . . . . . . . . . . . . 201


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13.2. Partial classical logic and the Lindenbaum property . . . . . 2 0 813.3. States on partial Boolean algebras . . . . . . . . . . . . . . . 2 10

13.4. The Lindenbaum property and the hidden variable problem . 214Chapter 14. Unsharp quantum logics . . . . . . . . . . . . . . . . . . 217

14.1. Paraconsistent quantum logic . . . . . . . . . . . . . . . . . . 21714.2. -Preclusivity spaces . . . . . . . . . . . . . . . . . . . . . . . 2 2014.3. An aside: similarities ofPQL and historiography . . . . . . 2 2 2

Chapter 15. The Brouwer Zadeh logics . . . . . . . . . . . . . . . . . 22515.1. The weak Brouwer Zadeh logic . . . . . . . . . . . . . . . . . 22515.2. The pair semantics and the strong Brouwer Zadeh logic . . . 22815.3. BZL3-effect realizations . . . . . . . . . . . . . . . . . . . . 234

Chapter 16. Partial quantum logics and Lukasiewicz quantum logic 237

16.1. Partial quantum logics . . . . . . . . . . . . . . . . . . . . . 23716.2. Lukasiewicz quantum logic . . . . . . . . . . . . . . . . . . . 2 4116.3. The intuitive meaning of the Lukasiewicz quantum logical

connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Chapter 17. Quantum computational logic . . . . . . . . . . . . . . . 2 4917.1. Quantum logical gates . . . . . . . . . . . . . . . . . . . . . . 25217.2. The probabilistic content of the quantum logical gates . . . . 25917.3. Quantum computational semantics . . . . . . . . . . . . . . . 2 62

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Synoptic tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Index of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Index of Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293


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List of Figures

1.1.1 The Benzene ring . . . . . . . . . . . . . . . . . . . . . 14

1.1.2 MO2: the smallest OML that is not a BA . . . . . . . . 151.1.3 The Greechie diagram of G12 . . . . . . . . . . . . . . . 171.1.4 The Hasse diagram of G12 . . . . . . . . . . . . . . . . . 18

1.4.1 Failure of bivalence in QT . . . . . . . . . . . . . . . . . 372.2.1 J18: the smallest OMP that is not an OML . . . . . . . 512.4.1 The Greechie diagram of GGM410 . . . . . . . . . . . . 592.4.2 The state s0 on GGM410 . . . . . . . . . . . . . . . . . . 605.1.1 WT: the smallest OA that is not an OMP . . . . . . . 915.3.1 M4: the smallest QMV that is not an MV . . . . . . . . 1065.4.1 The operation ofMwl . . . . . . . . . . . . . . . . . . 1085.4.2 The Hasse diagram of Mwl . . . . . . . . . . . . . . . . 1096.5.1 The Greechie diagram of

G52 . . . . . . . . . . . . . . . 1 34

6.5.2 The Greechie diagram of G58 . . . . . . . . . . . . . . . 1 359.1.1 Quasi-model for in R2 . . . . . . . . . . . . . . . . . . 173

11.3.1 The Greechie diagram ofG30 . . . . . . . . . . . . . . . 1 9017.1.1 A noncontinuous fuzzy square root of the negation . . . 260

ix


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List of Tables

1 The quantum structures . . . . . . . . . . . . . . . . . . 271

2 The labyrinth of quantum logics . . . . . . . . . . . . . 273

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Preface

The term quantum logic has entered our languages as a synonymfor something that doesnt make sense to our everyday rationality. Or,somewhat more technically but still in the common literature, it signifiessome generic sort of mystification of classical logic understood only by theilluminati. In the technical literature, it is most frequently used to designate

the set of projections (H) on a Hilbert space H or the set of positiveoperators E(H) which fall between the smallest and the largest projectionson H in a suitable ordering - or some algebraic generalization of one of these.

Thus, we have two concrete or standard quantum logics. These are struc-tures closely related to the usual mathematical formalism that underlies thefoundation of quantum theory (QT). The set (H) is the basis for the sharptheory and E(H) for the unsharp theory, in much the same way that classi-cal logic is based, in its sharp and unsharp manifestations, on (subalgebrasof products of) the two-element set {0,1} and the real unit interval [0,1],respectively.

There is a still unfolding panorama of structures that generalize thesetwo standard models. The theory of the foundations of quantum mechanics

called quantum logic studies the standard models and their abstractions.Confusion has persisted as to just what quantum logic is and how it

should be construed as a veritable logic. The purpose of this book is todelineate (what we know of) the quantum logics, to explain of what thepanorama of quantum logics consists and to present actual logics whose al-gebraic or Kripkean semantics are based on the algebraic models that arehistorically referred to as quantum logics. Our position is that there is notone but that there are many quantum logics. These logics have various mod-els, usually one of the orthomodular structures, which include orthomodularlattices, orthomodular posets, orthoalgebras, and effect algebras.

We present sufficient historical background to give the reader an idea ofhow the theory developed. However, far more is presented than is needed

to simply develop the logic, so that the novice may pick up the motivat-ing aspects of the subject. Readers not interested in the technical logicaldetails of quantum logics may gain an accounting of the mathematical as-pects of quantum logic, the models of the theory and how they relateto one another, by reading only the initial chapters. Readers wanting tolearn more about these algebraic structures are referred to (Kalmbach, 1983;Dvurecenskij and Pulmannova, 2000).

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Even the purist (non-quantum) logician may have some difficulty read-ing the technical logic in the second half of the book without the earlier

preparation; such readers could, however, begin in the logical sections andrefer to the earlier sections as needed. Quantum logicians, however, mayproceed directly to the latter chapters.

We are writing for a multidisciplinary audience, and we warn the readerthat we at times are too verbose for mathematicians, too pedantic for physi-cists, too glib for logicians, and too technical for philosophers of science.We assure the reader that we have had the whole readership in mind as wemade our compromises and we beg her indulgence.

Here is an outline of the organization of the book. There are two parts.Part I, which consists of the first seven chapters, presents the historical

background and the algebraic developments that motivate the syntax andunderlie the semantics of the sequel. Part II studies a variety of quantumlogics; in this Part, the term logic is used in the traditional sense, as atheory for a consequence relation that may hold between well-formed formu-las of a given language. Semantical characterizations are introduced, bothalgebraic semantics and Kripkean semantics.

We set the stage in Chapter 1 by presenting some abstract notionsneeded later and by sketching the historical underpinnings of the subject.We present some of the basic notions of ordered sets followed by a quickreview of Hilbert space and operators thereon. We recall von Neumannsaxioms for quantum theory and Birkhoff and von Neumanns inseminal ideathat propositions about quantum systems can be viewed as forming a kind

of logic more appropriately modeled by projections on a Hilbert space thanby a Boolean algebra. Then, we present von Neumanns axiomatic schemewhich is followed by a discussion of Mackeys treatment and the abstractionsthat derived from it.

Chapter 2 presents the abstract axiomatization of sharp quantum theory,as well as the notion of event. The events, it is argued, band together toform a -complete orthomodular poset. The abstract notion of state andobservable are given, and the previously developed structure is rephrased inan equivalent way as an event-state system. The interplay between eventsand states is further developed in preclusivity spaces and similarity spaces.A preclusivity space is a graph (X, ), where the relation is symmetricand irreflexive; the corresponding similarity space is the pair (X,

R), where

R = X X . Preclusivity spaces invariably induce a closure operatorY Y, for Y X, and the family of all closed sets Y : Y = Yforms a complete ortholattice.

The notion of the Yes-set, Yes(E), associated with each event E isintroduced. Yes(E) is the set of all states s with s(E) = 1. By regarding


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states as possible worlds, this notion later allows us to introduce a Kripkeansemantics.

The main purpose of Chapter 3 is to introduce and characterize the com-plete ortholattice (H) of all projections of a Hilbert space. We are led toview (H) and the states Sthereon as an event-state system, appropriatelybaptized the Hilbert event-state system. The basic lattice theoretic prop-erties of (H) are given, and we sketch the development of Piron, Keller,Soler, Holland, and Morash which some regard as the crowning mathemati-cal achievement of the quantum logic approach to the foundations of quan-tum mechanics, namely the characterization of (H) by lattice theoreticproperties.

In Chapters 4 and 5, we introduce E(H), the (possibly) unsharp exten-sion of (H). The underlying set of E(H) is the set of all the boundedoperators trapped between |O and 1I in the usual ordering of self-adjoint

operators. It forms an effect algebra, a type of structure that is then studiedin some detail, and it is called the standard unsharp effect algebra. Alongthe way, we introduce BZ-posets and effect BZ-posets, the Mac Neille com-pletion, and unsharp preclusivity spaces. The BZ-effect algebras admit 2complements, a Brouwer complement and the fuzzy complement of the ef-fect algebra. Having both allows us to define the necessity and possibilityoperators.

The Lukasiewicz operations of disjunction and conjunction are intro-duced and discussed. Axioms for MV algebras are given. These providean adequate semantic characterization for Lukasiewicz many valued logics.The quantum version follows, which consists of the axioms for what arecalled QMV algebras. The QMV algebras that correspond to effect algebras

are determined; they are called quasi-linear QMV algebras.Chapter 6 presents the corresponding abstract axiomatics for unsharp

QT. Chapter 7 investigates alternative notions for sharpness and the con-nections between them.

We begin Part II with a discussion of the nature of algebraic semanticsand Kripkean semantics. We give the algebraic characterization and theKripkean characterization for classical logic (CL) as well as for intuitionis-tic logic (IL), the former being provided by the Boolean algebras and thelatter by Heyting algebras. We review the notions of truth, logical truth,consequence, and logical consequence in both CL and IL; and we recall thatthe algebraic semantics and the Kripkean semantics of CL characterize thesame logic, i.e., the notion of logical truth (resp., logical consequence), for

the algebraic semantics is equivalent to that of the Kripkean semantics; thesame is true for IL.

In Chapter 8, we proceed to sharp quantum logic (QL). There are two:OL and OQL. In OL (resp., OQL), the algebraic semantics are classifiedby the class of all algebraic realizations based on ortholattices (resp., ortho-modular lattices). The similarity spaces discussed earlier provide the basisfor the Kripkean semantics ofOL. Although the details of the constructions


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are not the same as for CL and IL, the algebraic semantics for OL (resp.,OQL) and the Kripkean semantics of OL (resp., OQL) characterize the

same logic. We then specify OQL to the Hilbert event-state systems to ob-tain the algebraic and Kripkean realizations for a quantum system S withassociated Hilbert space H.

We then turn to a discussion of polynomial implications in OQL, arguingthat there is a favored one, the so-called Sasaki hook. This conditional maybe interpreted as a counterfactual conditional.

In Chapter 9, we introduce the notion of a quasi-model in both the al-gebraic and the Kripkean semantics ofQL. This notion allows us to discussthree interesting metalogical properties, called Herbrand-Tarski, Verifiabil-ity, and Lindenbaum, whose failure in QL represents a significant anomalousfeature of quantum logics.

Chapter 10 addresses syntax. In it we provide an axiomatization ofQL

in the natural deduction style, proving soundness and completeness theoremswith respect to the Kripkean semantics presented earlier. OQL, but not OL,admits a material conditional, and we prove a deduction theorem for OQL.

In Chapter 11, we prove that orthomodularity is not a first-order prop-erty. We also introduce Hilbert quantum logic (HQL) which is the logicthat is semantically based on the class of all Hilbert lattices. We argue thatHQL is strictly stronger than OQL.

In Chapter 12, we extend sentential quantum logic presenting an alge-braic realization and a Kripkean realization for (first-order) QL with itsattendant notions of satisfaction, verification, truth, logical truth, conse-quence in a realization, logical consequence, and rules of inference. Webriefly discuss quantum set theory in which the usual Boolean-valued mod-

els are replaced by complete orthomodular lattices, perhaps the most strik-ing feature of which is the failure of the Leibniz-substitutivity principle.Other approaches to quantum set theory are discussed, where the notion ofextensional equality is not characterized by membership. Not only the ex-tensionality axiom but Leibnizs principle of indiscernibles may be violated.

Proceeding from the total logics, those which are syntactically closed un-der the logical connectives and in which any sentence always has a meaningin both the algebraic and the Kripkean semantics, we present in Chapter 13partial classical logic (PaCL) in which sentences need not have a meaning.An interesting feature of PaCL is the following: some characteristic clas-sical laws (like distributivity) that are violated in standard quantum logicrepresent logical truths of PaCL.

After presenting an axiomatization ofPaCL and proving soundness andcompleteness theorems, we discuss the failure of the Lindenbaum property.We introduce a relativization (to some nonempty class of realizations) ofthis notion and relate this relativized notion to the relative validity of thelogical truths of classical logic as well as to the question of (non-contextual)hidden variable theory.


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In Chapter 14, we turn our attention to unsharp quantum logics. Insharp logics both the logical and the semantic version of the noncontradic-

tion principle hold. This means that is always false (in PaCL, whendefined), and it never happens that both and are true. In unsharp log-ics both conditions fail; thus, these logics are the natural logical abstractionsof the effect-state systems studied in Part I.

Paraconsistent quantum logic (PQL) is presented first. This logic hasan algebraic semantics based on bounded involution lattices and a Kripkeansemantics in which the accessibility relation is symmetric but not necessar-ily reflexive, while the propositions behave as in the OL case. The othersemantic definitions agree with OL, mutatis mutandis, and the algebraicand Kripkean semantics characterize the same logic. The axiomatization ofPQL looks like that of the OL calculus except that the absurdity rule andthe Duns Scotus rule are lacking. As in OL, the logic PQL satisfies the

finite model property and is therefore decidable.Regular bounded involution lattices give rise to a specialization ofPQL,

naturally called regular paraconsistent quantum logic (RPQL). The Krip-kean realizations of RPQL may be represented by -preclusivity spaces inwhich the preclusivity relation is sensitive only to a characteristic approxi-mation degree . The realizations are sharp when 0 < 12 and unsharpwhen 12 1.

In Chapter 15, we turn to a stronger unsharp quantum logic, the BrouwerZadeh logics. These are natural abstractions of the class of all BZ-latticesstudied in Chapter 4.

In Chapter 16, we study partial quantum logics (PaQL). There are threetypes: unsharp (UPaQL), weak (WPaQL) and strong (SPaQL). They

admit a semantic characterization corresponding to the class of all effectalgebras, orthoalgebras and orthomodular posets, respectively.

The language of PaQL consists of a set of atomic sentences and of twoprimitive connectives, the negation and the exclusive disjunction + (aut),while the conjunction is metalinguistically defined, via the de Morgan law.Whereas all disjunctions and conjunctions are considered legitimate froma linguistic point of view, a disjunction + will have the intended meaningonly in case the values of and are orthogonal in the corresponding effectalgebra, and when this is not the case an arbitrary meaning is assigned.

First, we give the semantics ofUPaQL, then the axiomatization which,unlike QL, admits only inferences from single sentences. Then, we add theDuns Scotus rule to get an axiomatization for WPaQL. Finally, we givean axiom which essentially states that the disjunction + behaves likea supremum whenever is orthogonal to , arriving at SPaQL. We thendiscuss soundness and completeness theorems.

We finish the chapter with a discussion of the Lukasiewicz quantum logic(LQL), which generalizes both OQL and L (Lukasiewicz infinite manyvalued logic). The semantics of LQL is based on QMV algebras.


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Finally, in Chapter 17, we discuss quantum computational logic. Thequantum generalization of a bit is called a quantum bit, or qubit; it is a unit

vector in the Hilbert space C2

. Quantum computations, or quantum logicalgates, are represented by unitary transformations on qubit systems. Fixingan orthonormal basis, called the computational basis, we define an n-qubitsystemto be an element of the n-fold tensor product nC2. Quregisters arequbits or n-qubit systems.

The quantum computational logic is based on certain quantum logicalgates defined on quregisters, the Toffoli gate, a reversible conjunction, nega-tion, disjunction, and the square root of negation. The latter is a genuinequantum gate in that it transforms classical registers into quregisters thatare superpositions.

The sentential language FormL is given by negation, conjunction andthe square root of negation, with disjunction then defined via the de Morgan

law. In quantum computational semantics, the meaning of the linguisticsentences are represented by quregisters.

A quantum computational realization of FormL is a function Qub fromthe set of all formulas in FormL to the union of the Hilbert spaces nC2.The image Qub() is called the information value of . Since Qub() isa unit vector of some Hilbert space, it has a probability-value which weuse to define a probability-value for itself. A sentence is true when itsprobability-value in a computational realization Qub is 1. The notions oflogical truth, consequence, and logical consequence are defined accordingly;the logic characterized by this semantics is called quantum computationallogic (QCL). Distinctions between QCL and QL are made. It is noted thatthe flavor of this logic is completely different from that of all the quantum

logics that preceded it.

Not addressed herein are the generalized sample spaces of the so calledAmherst School, initiated by Foulis and Randall (Foulis, 1999). This theoryattempts to generate a logic for any empirical situation, the models for whichare precisely the models that we develop for quantum logic. Much of thetheory of the algebraic structures that shall concern us was generated bythis School.

Nor have we addressed the generalized measure theory that has arisenin which a Boolean algebra of measurable sets is replaced by a more generalsuitable structure - one of the models that we discuss. While this develop-ment may turn out to be of mathematical importance in applications of thetheory, it does not directly impact the logic. Nor do we speculate on antic-ipated applications to the logical underpinnings of other empirical sciences,including biology, economics, political science, and psychology.

What we present is the best that we can do at the moment. We do notpresent the final definitive way of thinking quantum logically. Although, in


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most cases, what we do present seems to us natural enough to be lasting.Whatever may come of quantum logics in the future, we hope that a reading

of this book will leave no question that quantum logics are (or can be madeinto) logics in the truest sense of the logical tradition. We hope that wehave conveyed the collective enthusiasm of so many of the researchers inthis rapidly developing field.

Errors

Errata may be reported by email to [email protected] .A list of errata will be maintained at the web site:http://web.tiscalinet.it/giuntini/rqt/errata.html


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Acknowledgments

Many ideas and results presented in this book are based on long andintense scientific collaborations with other scholars, whose work seems tobe deeply entangled with our own. We would like to mention in primisG. Cattaneo, D. Foulis, S. Gudder, A. Leporati, R. Leporini, P. Mittelstaedt,and S. Pulmannova. All of them are coauthors of many joint articles that

have provided momentous inspiration for this book.Our research has also been greatly enhanced by many stimulating inter-actions with D. Aerts, E. Beltrametti, P. Busch, J. Butterfield, G. Cassinelli,E. Castellani, A. Dvurecenskij, C. Garola, G. Ghirardi, H. Greuling, P.Lahti, F. Laudisa, D. Mundici, M. Navara, P. Ptak, J. Pykacz, M. Redei,B. Riecan, Z. Riecanova, F. Schroeck, K. Svozil, G. Toraldo di Francia, andwith many members of the International Quantum Structure Association(IQSA) whose periodical meetings have represented splendid and fruitfuloccasions for scientific collaboration. Some great figures of the quantumstructure community are no longer with us; we take this occasion to re-member fondly M.K. Bennett, G. Birkhoff, S. Bugajski, C. Randall, G.Ruttimann.

Finally, we would like to thank warmly F. Paoli, who has carefully readthe manuscript and offered precious suggestions and assistance. Our gratefulacknowledgement also goes to R. Wojcicki and to the editors of Kluwerwho have constantly encouraged our work and patiently waited for the finalversion of this book.

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PART I Mathematical and Physical Background


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Introduction

In 1920 Lukasiewicz published a two-page article whose title was Onthree-valued logic. The paper proposes a semantic characterization for thelogic that has been later called L3 (Lukasiewicz three-valued logic). Inspite of the shortness of the paper, all the important points concerning thesemantics of L3 are already there and can be naturally generalized to the

case of a generic number n of truth-values as well as to the case of infinitemany values. The conclusion of the article is quite interesting:

The present author is of the opinion that three-valued logichas above all theoretical importance as an endeavour to con-struct a system of non-aristotelian logic. Whether the newsystem of logic has any practical importance will be seen onlywhen the logical phenomena, especially those in the deduc-tive sciences, are thoroughly examined, and when the conse-quences of the indeterministic philosophy, which is the meta-physical substratum of the new logic, can be compared withempirical data (Lukasiewicz, 1970c).

These days, Lukasiewicz remark appears to be highly prophetic, at leastin two respects. First of all, the practical importance of many-valued logicshas gone beyond all reasonable expectations at Lukasiewicz times. What wecall today fuzzy logics (natural developments of Lukasiewicz many-valuedlogics) gave rise to a number of technological applications. We need onlyrecall that we can buy washing machines and cameras whose suggestivename is just fuzzy logic.

At the same time, quantum theory (QT) has permitted us to comparethe consequences of an indeterministic philosophy with empirical data. Thishas been done both at a logico-mathematical level and at an experimentallevel. The so called no go theorems 1 speak to the impossibility of deter-ministic completions of orthodox QT by means of hidden variable theories .

Interestingly enough, some experiments that have been performed in theEighties2 have confirmed the statistical predictions of QT, against the pre-dictions of the most significant hidden variable theories.

1See (von Neumann, 1932; Jauch, 1968; Kochen and Specker, 1967; Bell, 1966; Giun-tini, 1991a).

2See (Aspect, Grangier and Roger, 1981; Aspect and Grangier, 1985).

3


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4 INTRODUCTION

Lukasiewicz was a contemporary of Heisenberg, Bohr, von Neumann.Strangely enough, however, he very rarely makes explicit references to QT.

In spite of this, he seems to be aware of the importance of QT for hisindeterministic philosophy. In 1946 he writes a revised version of his paperOn Determinism, an address that he delivered as the rector of the WarsawUniversity for the inauguration of the academic year 1922/1923. At the verybeginning of the article he notices:

At the time when I gave my address those facts and theoriesin the field of atomic physics which subsequently led to theundermining of determinism were still unknown. In ordernot to deviate too much from, and not to interfere with,the original content of the address, I have not amplified myarticle with arguments drawn from this branch of knowledge(Lukasiewicz, 1970b).

For Lukasiewicz, the basic reason that suggested going beyond classicalbivalent semantics was a philosophical one.3 His main argument, developedin the paper On Determinism can be sketched as follows:

First statement: bivalence implies determinism. Second statement: determinism contradicts our basic intuition

about necessity and possibility. Conclusion: bivalence has to be refused.

The argument essentially refers to temporal sentences that describe fu-ture contingent events . A typical example is represented by a sentence like

John will not be at home tomorrow noon.

More formally we can write a temporal sentence as: (t), to be read as

is the case at time t. A temporal sentence (t) may be true or false withrespect to a given time t1, which may either precede or follow t. We willwrite (according to the usual semantic notation): |=t1 (t) and |=t1 (t),respectively.

How can the first statement (bivalence implies determinism) be de-fended? Let us assume the bivalence principle:

any sentence is either true or false (tertium non datur).

And let us apply this principle to the case of temporal sentences. Supposet1 < t. By bivalence we have:

|=t1 (t) or |=t1 (t).

In other words, the event described by (t) is already determined at timet1. Suppose: |=t1 (t). Then, (t) turns out to describe a necessary event.As a consequence, we can conclude that: bivalence implies determinism.

How can we justify the second statement (determinism contradicts ourbasic intuition about necessity and possibility)? Suppose (t) describes acontingent event (such as the fact asserted by the sentence John will not

3See (Dalla Chiara and Giuntini, 1999).


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INTRODUCTION 5

be at home tomorrow noon). Now our basic intuition about contingencyand necessity seems to require that both sentences (t) and (t) must bepossible at time t1. As a consequence, future contingent events cannot bedetermined at any previous time (by definition of contingency).

Strangely enough, from the historical point of view, the abstract re-searches on fuzzy structures and on quantum structures have undergonequite independent developments for many decades during the 20-th century.

Only after the Eighties, there emerged an interesting convergence be-tween the investigations about fuzzy and quantum structures, in the frame-work of the so called unsharp approach to quantum theory. In this connectiona significant conjecture has been proposed: perhaps some apparent myster-ies of the quantum world should be described as special cases of some moregeneral fuzzy phenomena, whose behavior has not yet been fully understood.

The ambiguities of the quantum world can be investigated at different

levels. The first level involves the essential indeterminism of QT. In orderto understand the origin of such indeterminism, from an intuitive point ofview, it will be expedient to follow the argument proposed by Birkhoff andvon Neumann in their celebrated article The logic of quantum mechanics(Birkhoff and von Neumann, 1936). At the very beginning of their paper,Birkhoff and von Neumann observe:

There is one concept which quantum theory shares alike withclassical mechanics and classical electrodynamics. This is theconcept of a mathematical phase-space. According to thisconcept, any physical system S is at each instant hypotheti-cally associated with a point in a fixed phase-space ; this

point is supposed to represent mathematically, the state ofS, and the state ofS is supposed to be ascertainable bymaximal observations.

Maximal pieces of information about physical systems are also called purestates. For instance, in classical particle mechanics, a pure state of a singleparticle can be represented by a sequence of six real numbers r1, . . . , r6,where the first three numbers correspond to the position-coordinates, whilethe last ones are the momentum-components.

As a consequence, the phase-space of a single particle system can beidentified with the set R6, consisting of all sextuples of real numbers. Sim-ilarly for the case of compound systems, consisting of a finite number n ofparticles.

Let us now consider an experimental proposition P about our system,asserting that a given physical quantity (also called observable) has a certainvalue (for instance: the value of position in the x-direction lies in a certaininterval). Such a proposition P will be naturally associated with a subsetX of our phase-space, consisting of all the pure states for which P holds.In other words, the subsets of seem to represent good mathematical rep-resentatives of experimental propositions. These subsets have been called


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6 INTRODUCTION

by Birkhoff and von Neumann physical qualities. According to alternativeterminologies, physical qualities are also currently called physical properties

or physical propositions or physical questions or physical events . At thislevel of analysis, we will simply say events. Of course, the correspondencebetween the set of all experimental propositions and the set of all events willbe many-to-one. When a pure state p belongs to an event X, we can say thatthe system in state p verifies both X and the corresponding experimentalproposition.

What about the structure of all events? As is well known, the powerset of any set gives rise a Boolean algebra. And also the set F() of allmeasurable subsets of (which is more tractable than the full power setof , from a measure-theoretic point of view) turns out to have a Booleanstructure. Hence, we may refer to the following Boolean algebra:

F() , , , c , 0 , 1 ,where:

1) , , c are, respectively, the operations intersection, union, relativecomplement;

2) 0 is the empty space, while 1 is the total space.

According to a standard interpretation, the operations , , c can benaturally regarded as a set-theoretic realization of the classical logical con-nectives and, or, not. As a consequence, we will obtain a classical semanticbehavior:

a state p verifies a conjunction X

Y iff p

X

Y iff p verifies

both members; p verifies a disjunction X Y iffp X Y iffp verifies at least one

member; p verifies a negation Xc iff p / X iff p does not verify X.

In such a framework, classical pure states turn out to satisfy an im-portant condition: they represent pieces of information (about the physicalsystem under investigation) that are at the same time maximal and logi-cally complete. They are maximal because they represent a maximum ofinformation that cannot be consistently extended to a richer knowledge inthe framework of the theory (even a hypothetical omniscient mind couldnot know more about the physical system in question). Furthermore, pure

states are logically complete in the following sense: they semantically decideany event. For any p and X,

p X or p Xc.The semantic excluded middle principle is satisfied.To what extent can such a picture be adequately extended to QT?

Birkhoff and von Neumann observe:


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INTRODUCTION 7

In quantum theory the points of correspond to the so calledwave-functions and hence is a ... function-space, usually

assumed to be Hilbert space.As opposed to classical mechanics, QT is essentially probabilistic. Gen-

erally, pure states only assign probability-values to quantum events. Thisis strongly connected with the uncertainty relations , which represent oneof the most significant dividing line between the classical and the quantumcase. Let represent a pure state (a wave-function) of a quantum systemand let P be an experimental proposition (for instance the spin-value inthe x-direction is up). The following cases are possible:

(i) assigns to P probability-value 1;(ii) assigns to P probability-value 0;

(iii) assigns to P a probability-value different from 1 and from 0.

In the first two cases, we will say that P is true (false) for the systemin state ; in the third case, P will be semantically indeterminate. Thisconstitutes the first level of ambiguity or fuzziness.

As a consequence, unlike classical mechanics, in QT pure states turnout to represent pieces of information that are at the same time maximaland logically incomplete. Such divergence between maximality and logicalcompleteness is the origin of most logical anomalies of the quantum world.

A second level of ambiguity is connected with a possibly fuzzy characterof the physical events that are investigated. We can try and illustrate thedifference between two fuzziness-levels by referring to a nonscientific ex-ample. Let us consider the two following sentences, which apparently haveno definite truth-value:

I) Hamlet is 1.70 meters tall;II) Brutus is an honourable man.

The semantic uncertainty involved in the first example seems to dependon the logical incompleteness of the individual concept associated to thename Hamlet. In other words, the property being 1.70 meters tall isa sharp property. However, our concept of Hamlet is not able to decidewhether such a property is satisfied or not. Unlike real persons, literarycharacters have a number of indeterminate properties. On the contrary,the semantic uncertainty involved in the second example, is mainly causedby the ambiguity of the concept honourable. What does it mean be-ing honourable? One need only recall how the ambiguity of the adjectivehonourable plays an important role in the famous Mark Antonys mono-logue in Shakespeares Julius Caesar. Now, orthodox QT generally takesinto consideration examples of the first kind (our first level of fuzziness):events are sharp, while all semantic uncertainties are due to the logical in-completeness of the individual concepts, that correspond to pure states ofquantum objects. A characteristic ofunsharp QT, instead, is to investigatealso examples of the second kind (second level of fuzziness).


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8 INTRODUCTION

In the following chapters (of Part I) we will try to understand how theessential indeterministic and ambiguous features of the quantum world are

connected with the deep mathematical and logical structures of QT. Wewill first present the axiomatic foundations of sharp QT (Chapters 1-3).The unsharp version of the theory will be developed in Chapters 4-7.


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CHAPTER 1

The mathematical scenario of quantum theory and

von Neumanns axiomatization

In his celebrated book Mathematische Grundlagen der Quantenmechanik(von Neumann, 1932),1 John von Neumann proposed an axiomatic versionof sharp QT. This theory is often referred to as orthodox quantum theory.

As happens in the case of other physical theories, a number of axioms of

QT concern the mathematical interpretation of some basic physical concepts:physical system, state of a physical system, observable, and event.

Orthodox QT is developed in the framework of a privileged mathemat-ical scenario, represented by particular kinds of abstract spaces that arecalled Hilbert spaces. Such spaces have for QT the same role that is playedby phase-spaces in the case of classical mechanics. Any physical system Sis associated to a particular Hilbert space H, which represents the mathe-matical environment ofS. As a consequence, all the basic physical notionsconcerning S have counterparts that live in H.

Before presenting the axioms of sharp QT, it will be expedient to re-call some fundamental mathematical definitions for the readers who are notfamiliar with abstract algebra and functional analysis.

1.1. Algebraic structures

Some very important characters of the quantum theoretical (and quan-tum logical) game are represented by particular algebraic structures, thatplay a relevant role both for sharp and unsharp QT. It is therefore expe-dient to recall the definitions of some basic algebraic concepts. These willaccompany us throughout this book.

Definition 1.1.1. PosetA partially ordered set (called also poset) is a structure

B = B , ,where: B (the support of the structure) is a nonempty set and is a partialorder relation on B. In other words, satisfies the following conditions forall a,b,c B:

(i) a a (reflexivity);(ii) a b and b a implies a = b (antisymmetry);

1See also the English version (von Neumann, 1996).

9


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10 1. THE MATHEMATICAL SCENARIO OF QUANTUM THEORY

(iii) a b and b c implies a c (transitivity).2Definition 1.1.2. Bounded poset

A bounded poset is a structure

B = B , , 0 , 1 ,where:

(i) B , is a poset;(ii) 0 and 1 are distinct3 special elements of B: the minimum and the

maximum with respect to . In other words, for all b B:0 b and b 1.

Definition 1.1.3. LatticeA lattice is a poset B = B , in which any pair of elements a, b has a meeta b (also called infimum) and a join a b (also called supremum) suchthat:

(i) a b a,b, and c B: c a, b implies c a b;(ii) a, b a b , and c B: a, b c implies a b c.

In any lattice the following condition holds:

a b iff a b = a iff a b = b.Alternatively, a lattice can be also defined as a structure

B = B , , ,where the meet and the join are binary operations satisfying the follow-ing conditions:

(L1) a a = a; a a = a (idempotence);(L2) a b = b a; a b = b a (commutativity);(L3) a (b c) = (a b) c ; a (b c) = (a b) c (associativity);(L4) a (a b) = a; a (a b) = a (absorption).

(In fact, the idempotence of and , (L1), follows easily from the ab-sorption condition, (L4).)

On a lattice B = B , , , a partial order relation can be defined interms of the meet :

a b iff a b = a.It follows that:

a b iff a b = b.Generally, when we know that we are discussing lattices, we will indicate a

lattice as a structureB = B , , .

2For the sake of simplicity, in the following, we will often indicate a structure by usingsimply the name of its support.

3 For our physical and semantic applications, it is expedient to refer to nondegeneratealgebraic structures. Hence, we always assume that the minimum and the maximumelement of a bounded structure are distinct.


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1.1. ALGEBRAIC STRUCTURES 11

But in some cases, we will prefer the poset notation:

B=

B ,

.

Suppose the elements of the lattice represent events (or properties orpropositions ). Then, according to a standard (although sometimes erro-neous) logical interpretation, the lattice operations and will correspondto the connectives conjunction (and) and disjunction (or).

Consider a lattice B = B , , and let X be any set of elements ofB. If existing, the infimum

X and the supremum

X are the elements

of B that satisfy the following conditions:

(ia) a X : X a;(ib) c B : a X[c a] implies c X;

(iia) a X : a X;(iib) c B : a X[a c] implies X c.On can show that, when they exist the infimum and the supremum are

unique. A lattice is complete iff for any set of elements X the infimum

Xand the supremum

X exist. A lattice is -complete iff for any countable

set of elements X the infimum

X and the supremum

X exist.In many interesting situations, a poset (or a lattice) may have a more

sophisticated structure: the poset is closed under a unary operation thatrepresents a weak form of logical negation. Such a finer structure is repre-sented by a bounded involution poset.

Definition 1.1.4. Bounded involution posetA bounded involution poset (BIP) is a structure B = B , , , 0 , 1 where:

(i) B , , 0 , 1 is a bounded poset;(ii) is a unary operation (called involution or generalized complement)that satisfies the following conditions:

(a) a = a (double negation);(b) a b implies b a (contraposition).

Suppose we have a bounded poset B = B , , , 0 , 1, where is aunary operation satisfying the double negation principle. Then, the follow-ing conditions are equivalent for B:

(1) a b implies b a;(2) a b = (a b) (when one side exists so does the other side);(3) a b = (a b) (when one side exists so does the other side).

Conditions (2) and (3) are usually called the de Morgan laws .The presence of a negation-operation permits us to define an orthogonal-

ity relation , that may hold between two elements of a bounded involutionposet.

Definition 1.1.5. OrthogonalityLet a and b belong to a bounded involution poset. The object a is orthogonalto the object b (indicated by a b) iff a b.4

4We find it convenient to refer to the elements of sets as objects.


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A set of elements S is called a pairwise orthogonal set iffa, b S such thata = b, a b.A maximal set of pairwise orthogonal elements is a set of pairwise orthog-onal elements that is not a proper subset of any set of pairwise orthogonalelements.

Let us interpret and as an implication-relation and a negation-operation, respectively. Then, the orthogonality relation turns out todescribe a logical incompatibility relation:

a is orthogonal to b iff a implies the negation of b.

When a is not orthogonal to b we will write:

a b.The orthogonality relation is sometimes also called preclusivity; while itsnegation is also called accessibility.Since, by definition of bounded involution poset, a b implies b a(contraposition) and a = a (double negation), one immediately obtainsthat is a symmetric relation:

a b implies b a.Notice that 0 0 and that is not necessarily irreflexive. It may

happen that an object a (different from the null object 0) is orthogonal toitself:

a a (because a a).In fact, an object may imply its own negation. Objects of this kind arecalled self-inconsistent. Suppose now we have two self-inconsistent objects

a and b, and let us ask whether in such a case a is necessarily orthogonal tob. Generally, the answer to this question is negative. There are examples ofbounded involution posets (the smallest of which has only 4 elements) suchthat for some objects a and b:

a a and b b and a b.In the particular case where the answer to this question is positive for

any pair of elements, we will call our bounded involution poset a Kleeneposet (or also a regular poset).

Definition 1.1.6. Kleene posetA bounded involution poset is a Kleene poset (or also a regular poset) iff itsatisfies the Kleene condition for any pair of elements a and b:

a a and b b implies a b.Definition 1.1.7. Bounded involution lattice

A bounded involution lattice (BIL) is a bounded involution poset that is alsoa lattice.

By Kleene lattice (or regular lattice), we will mean a Kleene poset thatis also a lattice.


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Generally, bounded involution lattices and Kleene lattices may violateboth the noncontradiction principle and the excluded middle. In other

words, it may happen that:a a = 0 and a a = 1.

According to our logical interpretation, this means that contradictionsare not necessarily false, while the disjunction between a proposition and itsnegation is not necessarily true. As a consequence, Kleene lattices representan adequate abstract tool that permits us to model unsharp and ambiguousconcrete situations. In such situations, the involution operation is oftencalled fuzzy complement. As we will see, Kleene posets and Kleene latticeswill play an important role for unsharp QT.

Sharp QT, instead, makes use of stronger algebraic structures, whereambiguous situations (which give rise to violations of the noncontradiction

principle) are forbidden. Important examples of sharp structures (whichsatisfy the noncontradiction principle) are represented by orthoposets andortholattices (Birkhoff, 1967; Kalmbach, 1983).

Definition 1.1.8. Orthoposet and ortholatticeAn orthoposet(OP) is a bounded involution poset B = B , , , 0 , 1 that

satisfies the conditions:

(i) a a = 0 (noncontradiction principle);(ii) a a = 1 (excluded middle principle).

An ortholattice (OL) is an orthoposet that is also a lattice.The involution operation of an orthoposet (ortholattice) is also called

orthocomplementation (or shortly orthocomplement).

A -orthocomplete orthoposet (-orthocomplete ortholattice) is an or-thoposet (ortholattice) B such that for any countable set {ai}iI of pairwiseorthogonal elements the supremum

{ai}iI exists in B.Another important category of lattices are the lattices that satisfy the

distributivity property.

Definition 1.1.9. Distributive latticeA lattice B = B , , is distributive iff the meet is distributed over thejoin and vice versa. In other words:

(i) a (b c) = (a b) (a c);(ii) a (b c) = (a b) (a c).

Distributive involution lattices are also called de Morgan lattices.

In this framework, classical Boolean algebras can be then defined asparticular examples of de Morgan lattices.

Definition 1.1.10. Boolean algebraA Boolean algebra (BA) is a structure

B = B , , , , 0 , 1that is at the same time an ortholattice and a de Morgan lattice.


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In other words, Boolean algebras are distributive ortholattices.As we will see, the possibility of violations of the distributivity relations

will represent one of the most significant formal properties of the abstractstructures arising in the framework of sharp QT. As a consequence, thedifferent forms of sharp quantum logic that we are going to study will turnout to be strongly nonBoolean. Unsharp QT is also nondistributive in allbut special cases; so it too is strongly nonBoolean.

At the same time, some important quantum structures do satisfy a signif-icant weakening of distributivity, that is represented by the orthomodularityproperty.

Two leading actors of the algebraic quantum play are represented byorthomodular posets and orthomodular lattices.

Definition 1.1.11. Orthomodular poset and orthomodular latticeAn orthomodular poset (OMP) is an orthoposet

B = B , , , 0 , 1that satisfies the following conditions:

(i) a, b B, a b implies a b B;(ii) a, b B, a b implies b = a (a b).

An orthomodular lattice (OML) is an orthomodular poset that is also alattice.

One can prove that an ortholattice is an orthomodular lattice iff it doesnot contain the Benzene ring, whose Hasse diagram is given in Figure1.1.1.5

1

b a

a b

0

cccc

cccc

ccc

ccccccccccc

Figure 1.1.1. The Benzene ring

5 Hasse diagrams are the usual way of representing finite posets and lattices. In aHasse diagram of a poset B all the elements of B are depicted and an ascending line, say,from a up to b indicates that b is immediately above a in the ordering, i.e., b > a and ifb > c a then c = a.


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Clearly, any distributive ortholattice, i.e., any Boolean algebra, is ortho-modular. The smallest orthomodular lattice which is not a Boolean algebra,

called MO2, is given in Figure 1.1.2.

1

a a b b

0

oooooo

oooooo

oooooo

cccc

cccc

ccc

yyyyyy

yyyyyy

yyyyyy

yyyyyyyyyyyyyyyyyy

ccccccccccc

oooooooooooooooooo

Figure 1.1.2. MO2: the smallest OML that is not a BA

Another important property that represents a weakening of distributiv-ity and a strengthening of orthomodularity is represented by modularity.

Definition 1.1.12. ModularityA lattice B is called modular iffa, b B,

a b implies c B[a (c b) = (a c) b].Every modular ortholattice is orthomodular, but not the other way

around. Furthermore, any distributive lattice is modular.

A bounded poset (lattice) B may contain some special elements, calledatoms, that play a significant role.Definition 1.1.13. Atom

An element b of B is called an atom of B iff b covers 0. In other words,b = 0 and c B: c b implies c = 0 or c = b.

Apparently, atoms are nonzero elements such that no other element liesbetween them and the lattice-minimum.

Definition 1.1.14. AtomicityA bounded poset B is atomic iff a B {0} there exists an atom b suchthat b a.

Of course, any finite bounded poset is atomic. At the same time,there are examples of infinite bounded posets that are atomless (and hencenonatomic), the real interval [0, 1] being the most familiar example.

It turns out that any atomic orthomodular lattice B is atomistic in thesense that any element can be represented as the supremum of a set ofatoms, i.e., for any element a there exists a set {bi}iI of atoms such thata =

{bi}iI.


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Definition 1.1.15. Covering propertyLet a, b be two elements of a lattice B. Recall that b covers a iffa b , a = b,and c B[a c b implies a = c or b = c].A lattice B satisfies the covering property iff a, b B: if a covers a b,then a b covers b.

It turns out that an atomic lattice B has the covering property iff forevery atom a of B and for every element b B such that a b = 0, theelement a b covers b.

One of the most significant quantum relations, compatibility, admits apurely algebraic definition.

Definition 1.1.16. CompatibilityLet B be an orthomodular lattice and let a and b be elements of B. Theelement a is called compatible with the element b iff

a = (a b) (a b).One can show that the compatibility relation is symmetric. Interestingly

enough, the proof uses the orthomodular property in an essential way.6

Clearly, if B is a Boolean algebra, then any element is compatible withany other element by distributivity.

One can prove that a, b are compatible in the orthomodular lattice B iffthe subalgebra ofB generated by {a, b} is Boolean.7

We will see later (Theorem 3.1.2) that this abstract notion of compat-ibility turns out to agree with the concrete notion of compatibility betweenobservables in the framework of Hilbert space QT.

Definition 1.1.17. IrreducibilityLet B be an orthomodular lattice. B is said to be irreducible iff

{a B : b B (a is compatible with b)} = {0, 1} .IfB is not irreducible, it is called reducible.

Reducible orthomodular lattices can be represented as the cartesianproduct of smaller orthomodular lattices.8 IfYand Zare two orthomod-ular lattices then their cartesian product may be made into an orthomodularlattice by defining all operations coordinatewise.

Let B be an orthomodular lattice, let a B, and let [0, a] := {b B :0 b a}. Each such interval [0, a] can be made into an orthomodular

6See, for instance, (Kalmbach, 1983).7 See (Kalmbach, 1983). A subalgebra of an orthomodular lattice B =

B , , , , 0 , 1 is a structure B = B , , , , 0 , 1 where: (i) B B;(ii) , , are the restrictions of , , to B; (iii) 0 = 0 and 1 = 1; (iv) B isan orthomodular lattice. The subalgebra of B generated by the elements a, b is the small-est subalgebra of B that contains a and b. The notion of subalgebra can be naturallygeneralized to other classes of structures.

8Recall that the cartesian product of two sets Y Z is the set {(y, z) : y Y and z Z} of all ordered pairs with the first element coming from Y and the second from Z.


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lattice by imposing on [0, a] the induced ordering [0,a] and the relativizedorthocomplementation : [0, a] [0, a]. These are defined as follows: forb, c [0, a], b [0,a] c iff b c in B, and b

:= b a. Doing so makes [0, a]an orthomodular lattice.

Moreover, in a reducible orthomodular lattice B, if an element a B iscompatible with all the elements of B, then so is a and B is isomorphic to9the cartesian product of the orthomodular lattices [0, a] [0, a], in whichcase we write B = [0, a] [0, a].

Definition 1.1.18. SeparabilityAn orthomodular lattice B is called separable iff every set of pairwise or-thogonal elements of B is countable.

A useful technique for visualizing orthomodular posets or lattices isbased on a notion that has come to be called a Greechie diagram.10 Thisparticular kind of hypergraph is essentially based on the fact that a finiteBoolean algebra is completely determined by its atoms. The Greechie dia-gram of, say, a finite orthomodular poset B consists of a pair (X, L) whereX is the set of all atoms of B and L is the family of all maximal pairwiseorthogonal subsets of X. Thus, L is in one-to-one correspondence with theset of all maximal Boolean subalgebras, or blocks , ofB.11

For example, the Greechie diagram pictured in Figure 1.1.3 represents

a

bc

d e

cccc

cccc

cccc

ccccc

cccc

cccc

Figure 1.1.3. The Greechie diagram ofG129This will be defined shortly; for now read, essentially the same as.10See (Kalmbach, 1983) and Section 6.5 where a more complete discussion is

presented.11

A Boolean subalgebra of an algebra B (on which there is a naturally defined orderingand an involution making it a bounded involution poset with bounds 0 and 1) is asubalgebra B1 of B in which, under the induced natural ordering and involution, B1 =B1 , , , , 0 , 1 is a Boolean algebra. (Our notion of a Boolean subalgebra of anorthomodular poset is rather general because the fundamental operations of the subalgebramay not be precisely the fundamental operations of the enveloping algebra. For example, is a fundamental operation of a Boolean subalgebra but not of an enveloping orthomodularposet.) A maximal Boolean subalgebra of B is a Boolean subalgebra of B which is not aproper subalgebra of any Boolean subalgebra of B.


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the orthomodular lattice G12, whose Hasse diagram is pictured in Figure1.1.4.

1

a b c d e

a b c d e

0

yyyyyyyyyyyyyyyyyy

ccccccccccc

oooooooooooooooooo

oooooooooooooooooo

ccccccccccc

yyyyyyyyyyyyyyyyyy

ccccccccccc

oooooooooooooooooo

ccccccccccc

yyyyyyyyyyyyyyyyyy

ccccccccccc

oooooooooooooooooo

ccccccccccc

yyyyyyyyyyyyyyyyyy

Figure 1.1.4. The Hasse diagram ofG12

In a Greechie diagram, the slope of the lines means nothing, some linesmay even be curves. It is instructive to observe that, because c and c arecompatible with all the elements of G12, G12 = {0, 1} MO2.

In orthomodular structures the involution is crucial and must beadded, as we did in Figure 1.1.4. The facts that orthomodular posets andlattices are the unions of their maximal Boolean subalgebras and that theinvolution is uniquely determined on a Boolean subalgebra underlies thestrength of Greechie diagrams. It allows us to depict the n atoms of a fi-nite Boolean subalgebra rather than all 2n of its elements. Moreover, if anatom a is in many blocks, it is listed only once but is on each of the linescorresponding to those blocks. The example given in Figure 2.4.1 later inthe text involves blocks of varying sizes, the largest being 27 - and there areseveral of those. It would be difficult to draw meaningful information fromthe Hasse diagram of that 410-element orthomodular poset.

In many algebraic and logical problems an important role is played byfilters and ideals , that are particular subsets of the support of a lattice.

Let B = B , , , 0 , 1 be a bounded lattice.

Definition 1.1.19. Filter and ideal(i) A filter ofB is a nonempty subset F of B such that

(a) a F b B: a b F;(b) a F b F: a b F.

(ii) An ideal ofB is a nonempty subset I of B such that(a) a F b B: a b I;(b) a I b I: a b I.


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A proper filter (ideal) ofB is a filter (ideal) ofB that does not coincidewith the support B.

A principalfilter (ideal) is a filter (ideal) ofB of the form [a, 1] = {x B : a x}([0, a] = {x B : x a}).Definition 1.1.20. Maximal filter and ideal

A maximal filter (maximal ideal) ofB is a proper filter (proper ideal) of Bthat is not properly included in any proper filter (proper ideal) of B.

Definition 1.1.21. Complete filter and idealA complete filter (complete ideal) of a bounded involution lattice B is a

filter (ideal) such that for any a B, the filter (ideal) contains either a ora.

Some important connections between structures of the same kind are

represented by homomorphisms, embeddings and isomorphisms.Consider a pair of similar structures:

B =

B , Rn11 , . . . , Rnii , o

m11 , . . . , o

mjj

and

B =

B , Rn11 , . . . , Rnii , o

m11 , . . . , o

mjj

.

In other words, for any k (1 k i), the two relations Rnkk and Rnkk aresupposed to have the same arity (number of arguments). Similarly, for anyk (1 k j), the two operations omkk and omkk have the same arity.

Definition 1.1.22. HomomorphismA homomorphism from B into B is a map

h : B Bthat preserves the relations and the operations. In other words:

(i) for any k (1 k i) and any x1, . . . , xnk B:Rnkk (x1, . . . , xnk ) implies R

nkk (h(x1), . . . , h(xnk ));

(ii) for any k (1 k j) and any x1, . . . , xmk B:h(omkk (x1, . . . , xmk )) = o

mkk (h(x1), . . . , h(xmk )).

Definition 1.1.23. Embeddings and Isomorphisms

(i) An embedding of B into B is a homomorphism h such that h isinjective and h1 Ran(h) (the inverse ofh restricted to the rangeof h) is an injective homomorphism from Ran(h) to Dom(h) (thedomain of h).In other words, for any x, y B: x = y iff h(x) = h(y), and bothh and h1 preserve existing relations and operations.

(ii) An isomorphism from B to B is an embedding h such that h issurjective . In other words, any element of B is the h-image of anelement of B.


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Two similar structures B and B are called isomorphic, written B = B,when there is an isomorphism from B to B. An embedding of one orthoposetor ortholattice into another is called an ortho-embedding.

As already mentioned, privileged examples of Boolean algebras are rep-resented by set-theoretic structures.

Example 1.1.24. Let U be any set. Consider the structure

B = B , , , , 0 , 1 ,where:

B is the power set P(U) ofU. In other words, X P(U) iff X U,i.e., any element of X is also an element of U;

is the set-theoretic intersection. In other words:X Y = X Y = {x : x X and x Y} ;

is the set-theoretic union. In other words:X Y = X Y = {x : x X or x Y} ;

is the relative complement with respect to U. In other words:X = Xc = U X = {x U : x / X} ;

0 is the empty set, while 1 is the total set U.The structure

Bturns out to be a Boolean algebra.

In many situations it is useful to consider set-theoretic Boolean algebraswhose support B corresponds to a convenient proper subset of the full powerset of a given set. In such cases, one usually speaks of fields of sets .


B = B , , , , 0 , 1 ,where:

B is a field of sets F(U) over U. In other words, F(U) is a set ofsubsets of U, such that F(U) contains , U, and is closed underthe set-theoretic operations intersection

, union

and relative

complement c; is the set-theoretic intersection ; is the set-theoretic union ; is the relative complement c; 0 is the empty set, while 1 is the total set U.

As with the preceding example, the structure B also turns out to be aBoolean algebra.


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From an intuitive point of view, the elements of a set-theoretic Booleanalgebra can be interpreted as extensional properties or concepts that are

crisp. These properties are extensional, because they are identified withparticular sets. For instance, the numerical property prime number isidentified with its extension: the infinite set {2, 3, 5, 7, 11, . . .} containingprecisely all the natural numbers that satisfy the definition of prime num-ber. What has been traditionally called the intension of a given propertyis here completely neglected. Furthermore, our properties are crisp, in thesense that the membership relation is always precisely determined:

X B x U : either x X or x X.Tertium non datur! Uncertain or ambiguous membership relations are notadmitted.

In many concrete situations, properties and concepts are accompanied

by some ambiguous and uncertain borders. Let us think of examples likebeautiful, honest, tall, and so on. In such cases, it seems reasonableto assume that the membership relation is only determined for some objects.Generally the question

does the object x satisfy the property X?

may have no definite answer.Situations of this kind have suggested the introduction of the notion of

fuzzy set, deeply investigated in a rich literature after the pioneering workof Zadeh (1965); this has represented a natural development of Lukasiewiczapproach to many-valued logics. From the intuitive point of view, fuzzy setscan be imagined as collections whose membership relation is not generally

precisely determined. In other words, the questionx X?

admits three possible answers: 1) Yes, 2) No; 3) Indeterminate.Technically, the description of fuzzy sets can be developed by using a

convenient generalization of the concept of characteristic function.

Definition 1.1.26. Classical characteristic functionLet U be any classical set and let X be any subset of U. The classicalcharacteristic function of X is the function X : U {0, 1} such that

x

U : X(x) =

1 if x X,

0 otherwise.From an intuitive point of view, the characteristic function of X can be

regarded as a kind of procedure that always gives an answer to the questiondoes an object x belong to the set X?. The answer will be 1, when theobject belongs to X; 0, otherwise. Since there is an obvious one-to-onecorrespondence between sets and characteristic functions, classical sets aresometimes simply identified with their characteristic functions.


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0 is the classical characteristic function of the empty set , while 1is the classical characteristic function of the total set U.

The structure B is a de Morgan lattice that is not Boolean.If we define f g iff f g = f, one can prove that for any fuzzy sets f

and g:

f g iff f g = g iffx U[f(x) g(x)].As happens in the Boolean case, in many fuzzy situations it is interesting

to consider a structure whose support B is a convenient subset of the set ofall possible fuzzy subsets of a given set.


B = B , , , , 0 , 1 ,where:

B is a set of fuzzy subsets of U that contains the classical charac-teristic functions of the total set U and of the empty set . Further-more, B is closed under the operations , , defined as follows,for any x U:

f(x) = 1 f(x);(f g)(x) = min(f(x), g(x)); (f g)(x) = max(f(x), g(x)).

, , are the operations defined above; 0 is the classical characteristic function of the empty set , while 1

is the classical characteristic function of the total set U.

The structure B is a bounded de Morgan lattice that is not Boolean, unless

U = .One immediately sees why fuzzy structures generally give rise to viola-

tions of the noncontradiction principle. Suppose that f is the fuzzy set suchthat for all objects x in U:

f(x) = 1/2.

This means that the membership relation to the fuzzy set f is completelyindeterminate for the object x. As a consequence, we will have:

(f f)(x) = min(f(x), 1 f(x)) = 1/2.Since 0(x) = 0, we will obtain:

f

f

= 0.

Such a totally ambiguous fuzzy set is often called the semitransparentfuzzy set and indicated by 12 1.

Standard fuzzy sets give rise to nonBoolean structures that are distribu-tive. As a consequence, one could say that they represent a kind of semi-classical structure. As we will see, important examples of nondistributivestructures emerge in the framework of (sharp and unsharp) quantum theoryand quantum logic.


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1.2. The geometry of quantum theory

Let us finally turn to the most peculiar characters of the mathematical

scenario of QT: the abstract objects that live in the framework of Hilbertspaces. But, before introducing Hilbert spaces, it is expedient to recall thedefinitions of group, division ring, vector space and pre-Hilbert space.

Definition 1.2.1. GroupA group is a structure G = G , + , , 0, where + is a binary operation, is a unary operation, 0 is a special element. The following conditions hold:

(i) G , + , 0 is a monoid. In other words,(a) the operation + is associative:

a + (b + c) = (a + b) + c;(b) 0 is the neutral element:

a + 0 = a;

(ii) a G, a is the inverse of a:a + (a) = 0.

An Abelian monoid (group) is a monoid (group) in which the operation+ is commutative: a + b = b + a. We abbreviate a + (b) by a b.

Definition 1.2.2. Partially ordered groupA partially ordered group (or, po-group) is a structure G , + , 0 , thatsatisfies the following conditions:

(i) G , + , 0 is a group;(ii) G , is a poset;

(iii) the group translations are isotone; in other words, a,b,c,d G:c

d implies a + c

a + d and c + b

d + b.

Definition 1.2.3. RingA ring is a structure D = D , + , , , 0 that satisfies the following condi-tions:

(i) D , + , 0 is an Abelian group;(ii) the operation is associative:

a (b c) = (a b) c;(iii) the operation distributes over + on both sides, i.e., a,b,c D:

(a) a (b + c) = (a b) + (a c);(b) (a + b) c = (a c) + (b c).

If there is an element 1 in D that is neutral for (i.e., ifD , , 1 is a monoid),then the ring is called a ring with unity.

A ring is trivial in case it has only one element, otherwise it is nontrivial.It is easy to see that a ring with unity is nontrivial iff 0 = 1.

A commutative ring is a ring in which the operation is commutative.Definition 1.2.4. Division ring

A division ring is a nontrivial ring D with unity such that any nonzeroelement is invertible; in other words, for any a D (a = 0), there is anelement b D such that a b = b a = 1.


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(v) | = | , where is the identity ifD = R, and the complexconjugation if D = C.

The inner product .|. permits one to generalize some geometrical no-tions of ordinary 3-dimensional spaces.

Definition 1.2.8. Norm of a vector

The norm of a vector is the number | 1/2.Note that the norm of any vector is a real number greater than or equal

to 0.A unit (or normalized) vector is a vector such that = 1.Two vectors , are called orthogonal iff | = 0.Definition 1.2.9. Orthonormal set of vectors

A set {i}iI of vectors is called orthonormal iff its elements are pairwiseorthogonal unit vectors. In other words:

(i) i, j I(i = j) : i|j = 0;(ii) i I : i = 1.

The norm . induces a metric d on the pre-Hilbert space V:d(, ) := .

We say that a sequence {i}iN of vectors in V converges in norm (orsimply converges) to a vector ofV iff limi d(i, ) = 0. In other words, > 0 n N k > n : d(k, ) < .

A Cauchy sequence is a sequence {i}iN of vectors in V such that > 0 n N h > n k > n : d(h, k) < .

It is easy to see that whenever a sequence{

i}iN of vectors in V con-

verges to a vector of V, then {i}iN is a Cauchy sequence. The crucialquestion is the converse one: which are the pre-Hilbert spaces in which everyCauchy sequence converges to an element in the space?

Definition 1.2.10. Metrically complete pre-Hilbert spaceA pre-Hilbert space Vwith inner product .|. is metrically complete withrespect to the metric d induced by .|. iff every Cauchy sequence of vectorsin V converges to a vector of V.

On this basis, we can finally define the notion of Hilbert space.

Definition 1.2.11. Hilbert spaceA Hilbert space is a metrically complete pre-Hilbert space.

A real (complex) Hilbert space is a Hilbert space whose division ring isR (C). The notion of pre-Hilbert space (Hilbert space) can be generalizedto the case where the division ring is represented by Q (the division ring ofall quaternions).

We will now define some basic Hilbert-space notions, which will play animportant role in the mathematical formalism of QT.

Consider a Hilbert space H over a division ring D.


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where Dom(A) (the domain of A) is a subset ofH.Definition 1.2.16. Densely defined operator

A densely defined operator ofH is an operator A that satisfies the followingcondition: R+ H Dom(A) [d(, ) < ], where d representsthe metric induced by .|..

Definition 1.2.17. Linear operatorA linear operator on H is an operator A that satisfies the following condi-tions:

(i) Dom(A) is a closed subspace of H;(ii) , Dom(A) a, b D : A(a + b) = aA + bA.

In other words, a characteristic of linear operators is preserving the linearcombinations.

Definition 1.2.18. Bounded operatorA linear operator A is called bounded iff there exists a positive real numbera such that H : A a.

Definition 1.2.19. Positive operatorA bounded operator A is called positive iff H : |A 0.

Definition 1.2.20. The adjoint operatorLet A be a densely defined linear operator of H. The adjoint of A is theunique operator A such that

Dom(A) Dom(A) : A| = |A .Notice that A is not necessarily densely defined.

Definition 1.2.21. Self-adjoint operatorA self-adjoint operator is a densely defined linear operator A such thatA = A.

If A is self-adjoint, then , Dom(A) : A| = |A.If A is self-adjoint and everywhere defined (i.e., Dom(A) = H), then A isbounded.

Definition 1.2.22. Projection operatorA projection operator is an everywhere defined self-adjoint operator P thatsatisfies the idempotence property: H : P = PP .There are two special projections |O and 1I called the zero (or null projection)and the identity projection which are defined as follows:

H,

|O = 0 and 1I = .

Any projection other than |O and 1I is called a nontrivial projection.

Thus, P is a projection operator if Dom(P) = H and P = P2 = P.The set of all projection operators will be indicated by (H).

One can prove that the set C(H) of all closed subspaces and the set (H)of all projections ofH are in one-to-one correspondence. As we will see, such


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1.2. THE GEOMETRY OF QUANTUM THEORY 29

correspondence will play a very important role in the algebraic and logicalstructures of QT.

Let X be a closed subspace of H. By the projection theorem12

everyvector H can be uniquely expressed as a linear combination 1 + 2,where 1 X and 2 is orthogonal to any vector ofX. Accordingly, we candefine an operator PX on H such that

H : PX = 1(in other words, PX transforms any vector into the X-component of)

It turns out that PX is a projection operator of H.Conversely, we can associate to any projection P its range,

XP = { : (P = )} ,

which turns out to be a closed subspace ofH.For any closed subspace X and for any projection P, the following con-ditions hold:

X(PX ) = X; P(XP) = P.

Definition 1.2.23. Projection-valued measureLet B(R) be the set of all Borel sets of real numbers. A projection-valuedmeasure (called also spectral measure) is a map

M : B(R) (H),that satisfies the following conditions:

M() =|

O; M(R) = 1I; for any countable set {i}iI of pairwise disjoint Borel-sets:

M(

{i}iI) =i

M(i),

where (in the infinite case) the series converges in the weak operatortopology13 of the set of all bounded operators of H.

One can prove that there exists a one-to-one correspondence between theset of all projection-valued measures and the set of all self-adjoint operatorsof

H. This is the content of the spectral theorem.14 We will indicate by AM

the self-adjoint operator associated with the projection-valued measure M;while MA will represent the projection-valued measure associated with theself-adjoint operator A.

12See, for instance, (Reed and Simon, 1972).13For the notion of weak operator topology, see (Reed and Simon, 1972).14See, for instance, (Beltrametti and Cassinelli, 1981).


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Definition 1.2.24. The trace functionalLet {i}iI be any orthonormal basis for H and let A be a positive operator.The trace of A (indicated by Tr(A)) is defined as follows:

Tr(A) :=i

i|Ai .

One can prove that the definition of Tr is independent of the choice ofthe basis.

For any positive operator A, there exists a unique positive operator Bsuch that: B2 = A. If A is a (not necessarily positive) bounded operator,then AA is positive. Let |A| be the unique positive operator such that|A|2 = AA. A bounded operator A is called a trace-class operator iffTr(|A|) < .

Definition 1.2.25. Density operatorA density operator is a positive, self-adjoint, trace-class operator such thatTr() = 1.

It is easy to see that, for any vector , the projection P[] onto the1-dimensional closed subspace [] is a density operator.

Definition 1.2.26. Unitary operatorA unitary operator is a linear operator U such that:

Dom(U) = H; UU = UU = 1I.

One can show that the unitary operators U are precisely the operators

that preserve the inner product. In other words, for any , H :| = U |U .

Any pair of Hilbert spaces H1 , H2 gives rise to a new Hilbert spaceH1 H2 that represents the tensor product of H1 and H2. As we will see,tensor products play an important role for the mathematical representationof compound quantum systems. They are also systematically used in themathematical formalism of quantum computation (see Chapter 17).

Definition 1.2.27. Tensor product Hilbert spaceLet H1 and H2 be two Hilbert spaces over the same field D (the real or thecomplex numbers). A Hilbert space

His the tensor product of

H1 and

H2

iff the following conditions are satisfied:(i) there exists a map (called tensor product) from the cartesian

product H1 H2 into H that satisfies the following conditions:(a) the tensor product is linear in each component; in other

words, , H1 , H2 a, b D:(a1) (a + b) = (a) + (b) ;(a2) (a + b) = (a) + (b);


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1.3. THE AXIOMATIZATION OF ORTHODOX QT 31

(b) the external product with a scalar carries across the tensorproduct; in other words, H1 H2 a D:a( ) = (a) = (a);(ii) every vector ofH can be expressed as a (Hilbert) linear combination

of vect

2004 reasoning in quantum theory sharp and unsharp quantum logics

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